Bernstein-Walsh theorems for harmonic functions in \(\mathbb{R}^n\) (Q2782558)

The classical Bernstein-Walsh theorem can be regarded as a quantitative version of Runge's theorem. It relates the rate of best polynomial approximation to a given function \(f\) on a (suitable) compact set in \(\mathbb{C}\) to the maximal domain of holomorphic extension of \(f\). Several authors have considered analogues for holomorphic functions in \(\mathbb{C}^n\) and for harmonic functions in \(\mathbb{R}^n\). The latter theory, which is less fully developed, is the subject of the present work. The paper presents a Bernstein-Walsh-type theorem for harmonic functions in \(\mathbb{R}^n\) using notions from ``\(Lh\)-potential theory'', which was previously introduced by the second author and is further elaborated here. The result seems to be necessarily rather complicated.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].